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Not in general -- there can be linkage disequilibrium among the loci. For instance, say that there are two di-allelic loci, $A/a$ and $B/b$, and that the frequencies of the $A$ and $B$ alleles are both $1/2$ and that they have the same effect on the trait, with no dominance. If all haplotypes in the population are either $Ab$ or $aB$ (with no $AB$ or $ab$ haplotypes), then the genetic variance at each locus is high, but the total genetic variance for the trait is 0!

Thanks for your answer. Yes that totally makes sense. So the formula would be $$\sum_{i=1}^{n} \left( \sigma_i \frac{\sum_{j=1}^{n} 1-\text{cor}(i,j)}{n} \right)$$ , where $\text{cor}(i,j)$ is the correlation between the two loci... or something like that…?
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Remi.bApr 12 '14 at 20:06